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I The cases in proving that group of order 90 is not simple

  1. Dec 8, 2018 at 4:55 PM #1
    https://imgur.com/a/FuCPJLe

    I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a contradiction, but why is the second case there? In other words, why do these two cases exhaust all possibilities?

    EDIT: Actually, I think that I see it now. Since the intersection of subgroups is a subgroup, by Lagrange we must have that the negation of two distinct Sylow 3-subgroups intersecting trivially is intersecting with 3 elements.
     
    Last edited: Dec 8, 2018 at 5:17 PM
  2. jcsd
  3. Dec 9, 2018 at 12:48 AM #2

    mathwonk

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